A person taught his son about sine and cosine. He himself got introduced to them as ratios of side lengths in a right triangle, but he didn't like the idea of changing definition when angles become more than 90 degrees, so he defined those functions as abscissa and ordinate of a point on a circle of unit radius, centered at origin.

I think this is not perfect. Education is more of "progressing towards lesser and lesser lies", and changing definitions is an important part. The student might face it when he'll wonder about equation sin x = 2 , which will get to complex numbers.

Similarly, here getting a one less power of x might correspond to "divide by x". But might sometimes not - choosing that it actually does correspond to "divide by x" is a choice. Often obvious, but sometimes not - which is seen in Gelfand's explanation of why "negative multiplied by negative makes positive", or similarly, why 0^0 is 1.

Just saying that "x to one lesser power is the same divided by x" can also be seen as a convention (e.g. for some objects division can be not defined). And if it's a convention, not universal truth... then to somebody who's studying the subject this convention should be justified.

Yes, but in my experience, it helps to thoroughly understand (down to first principles if you want to), and then memorize anyway.

I quickly figured out that even if I've deeply spent time with a subject, understanding every step and derivation of some equation, if I can just quickly pop up equations (and other facts) in my head to "look" at them, it not only helps with application, but also with further understanding.

Being able to quickly recite the Taylor Series or an Inverse Fourier Transform in my head to apply in a problem beats stuff like "oh I remember understanding how it was derived, but I'd need to look it up", because all the details I otherwise once understood but did not bother memorizing might be important.

x⁰ is generally a matter of definition and not a fact reasonably accessed from deeper underlying fundamentals. It just so happens that the definition fits this story that you have for reasons of convenience. Also, you know, 0⁰.